An allusion is defined as any reference to another literary work, figure, or historical event. Given the context of Zusak's novel, set as it is in Germany during the Second World War, the majority of the allusions refer to real life characters and events, such as Hitler and the Munich Olympic Games. The one text that is alluded to repeatedly is Mein Kampf, Hitler's own book translated as "My Struggle." Note how Hans Huberman in Part 3 gets hold of this book:
He knocked on the door of the Nazi Party office in Molching and took the opportunity to ask about his membership application. Once this was discussed, he proceded to give them his last scraps of money and a dozen cigarettes. In return, he received a used copy of Mein Kampf.
Such allusions become incredibly important to the novel, as remember it is the pages of Mein Kampf that Max paints white and blanks out so that he can populate them with his own story, which allows him to recreate reality and present an alternative version of events where he triumphs over Hitler in a boxing match. The repeated historical allusions in this novel therefore serve to establish the context in which it is set and are used by the author to show how the characters he has created fight against that context to imagine alternative realities and dream different lives for themselves.
As the other educator has correctly mentioned, allusions are indirect references to people, places, and things of historical, political, or social significance.
In Part Three, an example of allusion is a name uttered by Arthur Berg, the fifteen-year-old leader of a fruit-stealing gang:
Arthur had moved on to Rudy. “And you’re the Jesse Owens one, aren’t you?”
Jesse Owens was an American runner known for his speed: he won four Gold medals at the 1936 Berlin Olympics in Germany. Through his amazing feats as an athlete, Jesse Owens, a black man, invalidated the basic tenets of Hitler's Aryan supremacist ideology. In the reference above, Rudy is said to be as fast as Jesse Owens. Liesel argues that Rudy is fast enough to be a part of Arthur Berg's fruit-stealing gang, alluding to Rudy's great speed.